Pure states in the SYK model

and nearly-AdS2 gravity

Ioanna Kourkoulou1 and Juan Maldacena2

1Jadwin Hall, Princeton University, Princeton, NJ 08540, USA

2Institute for Advanced Study, Princeton, NJ 08540, USA

Abstract

We consider pure states in the SYK model. These are given by a simple localcondition on the Majorana fermions, evolved over an interval in Euclidean time toproject on to low energy states. We find that “diagonal” correlators are exactly thesame as thermal correlators at leading orders in the large N expansion. We alsodescribe “off diagonal” correlators that decay in time, and are given simply in termsof thermal correlators. We also solved the model numerically for low values of Nand noticed that subsystems become typically entangled after an interaction time.In addition, we identified configurations in two dimensional nearly-AdS2 gravity withsimilar symmetries. These gravity configurations correspond to states with regionsbehind horizons. The region behind the horizon can be made accessible by modifyingthe Hamiltonian of the boundary theory using the the knowledge of the particularmicrostate. The set of microstates in the SYK theory with these properties generatesthe full Hilbert space.

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Contents

1 Introduction 1

2 Definition of the model and the initial states 2

3 The large N solution 43.1 Two point functions from thermal ones . . . . . . . . . . . . . . . . . . . . 43.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Low energy limit , almost conformal limit 7

5 Some exact diagonalization results 95.1 The pure states as a typical state in Hilbert space . . . . . . . . . . . . . . 95.2 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.3 Entanglement entropy of subsystems . . . . . . . . . . . . . . . . . . . . . 105.4 Euclidean correlators at finite N . . . . . . . . . . . . . . . . . . . . . . . 11

6 Gravity interpretation 126.1 Qualitative connection with other boundary state solutions . . . . . . . . 15

7 Evolution of the state under a different Hamiltonian 187.1 Adding particles behind the horizon . . . . . . . . . . . . . . . . . . . . . 207.2 Relation to traversable wormholes . . . . . . . . . . . . . . . . . . . . . . . 21

8 Discussion 21

1 Introduction

The SYK model involves N Majorana fermions undergoing few body interactions withrandom couplings [1, 2]. At low energies, it is a maximally chaotic model that has somefeatures in common with near extremal black holes, or more precisely, with nearly-AdS2

gravity [3, 2, 4, 5, 6, 7]. In this paper, we consider the evolution of particular pure statesin the SYK model. We study some aspects of the thermalization of these states. We alsoattempt to draw some lessons for the geometry associated to particular microstates fornearly-AdS2 black holes.

The particular initial pure states are obtained by combining pairs of Majorana fermionsinto qubit like operators and choosing states with definite eigenvalues for the σ3 compo-nents of all qubits. By choosing different eigenvalues we get a whole basis of the Hilbertspace. We further evolve these states over some distance ` in Euclidean time in order toget low energy states.

Up to the first few orders in the 1/N expansion, one can compute the correlators of thismodel in terms of finite temperature correlation functions, with β = 2`. The correlators of

1

fermions with the same index turn out to be the same as thermal correlators. We interpretthis as saying that the “gravitational background” for these states is the same as that ofthe thermal state. Some other correlators, such as correlators involving fermions withdifferent indices, are different from the thermal ones, which are zero for different indices.However, they can still be computed in terms of suitable thermal correlators. These aresuch that they decay away under Lorentzian evolution, reflecting the thermalization of thesystem.

The gravity dual of the SYK model is not precisely known. However, at low energiesthere is an emergent reparametrization symmetry that is both spontaneously and explicitlybroken [2]. This pattern of symmetries is also present in nearly-AdS2 gravity, wherethe reparametrization symmetry is the asymptotic symmetry of AdS2 [4, 5, 6, 7]. Inthe same spirit, we identify some nearly-AdS2 gravity configurations that have propertiessimilar to the pure states in the SYK model. Namely, we will see that the symmetries arebroken in a similar fashion. These pure states have a gravity description which involvesagain the full AdS2 space, but we introduce a shockwave at tL = 0 on the left boundary.Correspondingly, in Euclidean space, we continue to have a disk, but with a special pointat the boundary. This special point is the source of the shock wave in the Lorentziangeometry. An interesting feature of the geometric configuration is that it contains a regionbehind the horizon. The shock wave is separated from the horizon. This suggests that wehave a whole basis of states in the Hilbert space with a smooth horizon. The region behindthe horizon is not accesible to simple experiments by the boundary observer. However, asstudied in [8], evolving the system with a modified Hamiltonian we can make some of theregion behind the horizon visible. Here, we can make the whole t = 0 spatial slice visible.

For this purpose he/she has to add a term to the Hamiltonian that depends on theparticular microstate that is chosen. The procedure is essentially the same as the one thatrenders wormholes traversable [9, 10].

This paper is organized as follows. In section two we define the model and define a setof simple initial states. In section three we display the large N solution. In section fourwe discuss some aspects of the low energy limit. In section five we present some numericaldiagonalization results, discussing the decay of correlators, the rise of entanglement en-tropy, and we check statements about the large N solution. In section 6 we discuss someaspects of the gravity interpretation. In section 7 we give the protocol for looking behindthe horizon for these states. We end with a final discussion.

2 Definition of the model and the initial states

We consider the SYK model [1, 2]. We consider a Hilbert space generated by an evennumber, N , of Majorana fermions ψi, with {ψi, ψj} = δij. It has a Hamiltonian of theform

H =∑

1≤i<k<l<m≤N

jiklmψiψkψlψm , with 〈j2

iklm〉 =3!J2

N3(2.1)

2

with couplings jiklm which are all independent random numbers drawn from a gaussiandistribution with variance set by J , see [11] for more details. More generally one can alsoconsider a model with a Hamiltonian involving q fermions H = i

q2

∑k1···kq jk1···kqψ

k1 · · ·ψkq .We are interested in considering the evolution of special pure states. For example, we

can consider the state |B〉 that obeys the conditions

(ψ1 − iψ2)|B〉 = 0 , (ψ2k−1 − iψ2k)|B〉 = 0 , k = 1, · · · , N/2 (2.2)

If we imagine ψ1 and ψ2 as proportional to the σ1 and σ2 Pauli matrices, then |B〉 will havespin minus under σ3. More precisely, we can say that the state |B〉 has all plus eigenvaluesfor the operators

Sk ≡ 2iψ2k−1ψ2k , S2k = 1 , k = 1, · · · , N/2 (2.3)

More generally, we can define a whole set of states |Bs〉 by the conditions

(ψ2k−1 − iskψ2k)|Bs〉 = 0 , or Sk|Bs〉 = sk|Bs〉 , with sk = ±1 (2.4)

This defines 2N/2 states, one for each choice of the signs of all the sk, which form a basisof the Hilbert space.

The SYK evolution with (2.1) will give states which are linear combinations of thesestates. At long times we expect to get fairly generic linear combinations so that the statebecomes effectively thermalized (even though it remains a pure state). If we think of |B〉as a simple state were all qubits point up, then the evolution will start flipping some of thequbits so that we start getting a more general superposition of states with qubits pointingup and down1. More importantly, we get linear superpositions of such states. The SYKevolution can be viewed as a set of simple quantum gates that acts on the simple stategenerating a more complex state.

We expect these states, (2.4), to have an energy close to zero for the SYK hamiltonian.These are states of high energy compared to the minimal energy of the SYK model whichis of order E0 = −(number)N .

We can produce lower energy states by evolving with the Euclidean Hamiltonian|B(`)〉 = e−`H |B〉. In this way we can form an overcomplete set of low energy states.We expect that the expansion of |B(`)〉 is in terms of the energy eigenstates is

|B(`)〉 ∼∑α

e−`Eαcα|Eα〉 (2.5)

where the typical |cα|2 is of order of 2−N2

+1 2. In figure 2 we see an example for N = 30obtained by exact numerical diagonalization.

1The SYK evolution does preserve the sign of∏N/2k=1 Sk = (−1)F .

2Of course, some of the cα are zero for symmetry reasons. For example, since (−1)F commutes withthe Hamiltonian, all states that appear have to have the same value of (−1)F as the state |B〉.

3

Note that we can average the correlators over all choices of signs sk. This reproducesthe thermal ensemble exactly,∑

sk

〈Bs(`)|ψ · · ·ψ|Bs(`)〉 = Tr[e−βHψ · · ·ψ] , β = 2` (2.6)

This means that we can view the states |Bsk(`)〉 as an (overcomplete) basis of the lowenergy states relevant to the dynamics at temperature β. We see that after averaging overall sign choices {sk} for the states |Bs(`)〉 we get the exact thermal average. Of course,(2.6) is not at all surprising, given the way we have defined the states. What is moreinteresting is that each individual state |Bs(`)〉 gives rise to correlators that look thermalto high accuracy, as we will demonstrate below.

These “boundary states” are the one dimensional analog of similar boundary states thatwere used in [12, 13] to model quenches from vacua of gapped short range Hamiltoniansto CFTs in 1+1 dimensions. Related quantum quenches in the SYK model are discussedin [14].

3 The large N solution

3.1 Two point functions from thermal ones

In this section we analyze this problem in the large N limit. The proper way to treat theaverage over random couplings is to introduce replicas [11]. However, one finds that (forthe replica diagonal solution) the interaction between replicas is down by 1/N3 (or 1/N q−1

more generally). This means that, to leading orders in N , we can treat the couplingsjiklm as time independent gaussian fields with the two point function in (2.1). In thisapproximation, the model has an O(N) symmetry. A particularly interesting subgroupof O(N) is the one that flips the sign of any of the even indexed fermions. For example,we can consider the element that flips the sign of ψ2 leaving the rest unchanged. Thereis another element that flips the sign of ψ4, etc. We call this collection of O(N) groupelements, the “Flip Group” (it is generated by reflections along the even directions). Theboundary states |Bs〉 (2.4) are not invariant under these elements. The element that flipsthe sign of ψ2, changes the sign of s1 in |Bs〉, so it maps one possible state into another. Wecan think of this as flipping the sign of the first spin. Notice that two point functions suchas ψ1(τ1)ψ1(τ2) or ψ2(τ1)ψ2(τ2) are individually invariant under the “Flip Group”. Wecall such two point function “diagonal” two point functions. Diagonal correlators have thesame values in all |Bs〉 states. On the other hand we have also shown that the average overall states |Bs〉 is the same as the thermal average (2.6). This means that these two pointfunctions have a value with is identical to their thermal averages. The same argument alsoimplies that the following overlap is given in terms of the partition function

〈Bs|e−2`H |Bs〉 = 2−N/2Z(β) , β = 2` (3.7)

4

since the same argument indicates that it is independent of sk, to leading order in the1/N q−1 expansion. Note that both sides of (3.7) also have a 1/N expansion. This equationsays that (2.6) holds to good approximation for each state, not just in average. This is thesame self averaging that we can invoque regarding the random couplings, but now appliedto the initial state3.

The operator ψ1(τ1)ψ2(τ2) is not invariant under a reflection of the 2̂ axis. However,we can consider the combination ψ1(τ1)ψ2(τ2)S1 which is indeed invariant. Note that∑

s

〈Bs|ψ1(τ1)ψ2(τ2)s1|Bs〉 =∑s

〈Bs|ψ1(τ1)ψ2(τ2)2iψ1(0)ψ2(0)|Bs〉 =

= 2iT r[e−2`Hψ1(τ1)ψ2(τ2)ψ1(0)ψ2(0)] ∼ −2iGβ(τ1)Gβ(τ2) (3.8)

Since ψ1(τ1)ψ2(τ2)S1 is invariant under the “Flip Group”, (3.8) also shows that the finalresult also holds over element of the first sum, for each state |Bs〉.

In conclusion, defining the two point functions in the state |Bs〉 as

Gdiag(τ, τ ′) =〈Bs(`)|ψi(τ1)ψi(τ2)|Bs(`)〉

〈Bs(`)|Bs(`)〉= , no sum (3.9)

=〈Bs|e−(2`−τ2)Hψie−(τ2−τ1)Hψie−τ1H |Bs〉

〈Bs|e−2`H |Bs〉, τ2 > τ1 , no sum

Goff(τ, τ ′) = sk〈Bs(`)|ψ2k−1(τ1)ψ2k(τ2)|Bs(`)〉

〈Bs(`)|Bs(`)〉, no sum (3.10)

we find that

Gdiag(τ, τ ′) = Gβ(τ − τ ′) , β = 2` (3.11)

Goff(τ, τ ′) = 2i〈ψ1(τ)ψ2(τ ′)ψ1(0)ψ2(0)〉β = −2iGβ(τ)Gβ(τ ′) + o(1/N) , (3.12)

These results are valid at leading order in the 1/N q−1 expansion, but the last equality in(3.12) is valid only to leading order in the 1/N expansion. In principle, it is possible toadd the first 1/N correction to (3.12) that comes from the connected part of the four pointfunction. Notice that part of the statement is that in (3.9) (3.10) there is no dependence oni or k or the set of sk. In the second line of (3.9) we have assumed that τ1 < τ2 (otherwiseit needs to be reordered).

3.2 Comments

Let us note the following points

• The diagonal correlators are exactly thermal at large N . In particular, they onlydepend on the difference of times, despite the presence of the boundary at τ = 0, 2`.

3These two self averages are related if we view the initial state as arising from a long Euclidean timeevolution with random cuadratic couplings.

5

B

B τ = 2 = βτ =

B

(a)

(d)

(b)

Euclidean evolution

evolution

Lorentzian

(c)

τ=0

τ

τ=0

O O1 2

B

B

B

τ = 2 = β

B( )

Figure 1: (a) The Euclidean computation pictured as an interval of size β = 2`. (b) Thesame Euclidean computation pictured as a circle, with a special point where we projecton to the state |B〉. (c) Evolving by Euclidean time ` we get the state |B(`)〉, which wecan then evolve using Lorentzian evolution. (d) By inserting simple operators O1, O2, atintermediate Euclidean times we can get states containing some small excitations.

If we know the numerical or analytic (e.g. at large q) large N solution of the finitetemperature model, then (3.11), (3.12) give us a direct solution for the pure stateproblem.

• Note that there is no singularity in Goff at τ = τ ′ since the two different fieldsanticommute.

• (3.11) (3.12) obey the boundary condition Gdiag(τ, 0) = −iGoff(τ, 0) implied by (2.2),after using the UV form of the thermal correlator, Gβ(0) = 1

2.

• Douglas Stanford has checked (3.11) (3.12) against a direct numerical solution of thelarge N Schwinger Dyson equations with the appropriate boundary conditions [15].

• The diagonal correlators are independent of the state we started from. They areindependent of the choice of sk in (2.4).

• The off diagonal correlators are non-zero and they depend on the precise initial statewe start from, through the signs sk.

• Notice that these formulas are valid for all values of βJ , or `J , to leading order inthe 1/N q−1 expansion (for βJ � N).

6

• For q ≥ 4, (3.11) at order 1/N implies that all the operators that appear in the OPEof ψiψi also have the expectation values as in the thermal state.

• If we interpret the diagonal correlators as giving rise to the full “gravity” background,then this background is exactly the same as the one we have for the thermal state,or the thermofield double.

• We can set τ = `+ it to get the Lorentzian correlators at time t in the state |B(`)〉.In particular, note that the thermal correlator Gβ(β

2+ it) is real, so that (3.12) is

consistent with the anti-hemiticity of the operator (3.10) when t = t′.

• We can view the Euclidean interval as a full circle with a point where the statesrunning along the circle are projected to joint eigenstates of the operators Sk, seefigure 1(b).

• By inserting operators at Euclidean times 0 < τi < `, we can get other “close-by”states, see figure 1(d).

• The classical action on the solution gives us the overlap (3.7), and it coincides withthe partition function up to a constant, as indicated in (3.7).

• The following is a side comment. One could imagine starting with the SYK modelin (2.1) and then attempt to introduce a time dependence by taking a Hamiltonianwhich is Hnew = g(t)H, where H is in (2.1). However, this time dependence canbe completely removed by redefining the time to t̃, via dt̃ = g(t)dt. In terms of thetime t̃ we have the standard time independent Hamiltonian. Of course, if we wereto change individual couplings relative to each other, that can change the physics.

• Suppose we define the ratio

〈Bs(`)|ψi(τ1)ψi(τ2)|Bs(`)〉Tr[e−βH ]

(3.13)

instead of the more natural one in (3.9). We can now argue that the average overcouplings of this new ratio gives exactly the same as the average over couplings ofthe thermal correlator. The reason is that when we compute this ratio using thereplica trick, we impose the |B〉 boundary condition on one replica but the thermalone on the rest. Then the same symmetry argument we used to get (3.7) is valid forthe replicated problem. We checked this for N = 24 in figure (5).

4 Low energy limit , almost conformal limit

In this section we will discuss some low energy aspects of the above formulas (3.11) (3.12).We consider 1 � `J � N so that we go to the almost conformal limit. In this case we

7

can use the conformal limit of the thermal correlators [1, 2, 11]

Gβ(τ) =c∆[

Jβπ

sin πτβ

]2∆, c∆ ≡

[(1

2−∆

)tanπ∆

π

]∆

(4.14)

where ∆ = 1/4 for the Hamiltonian in (2.1)4. We get the off diagonal correlator inLorentzian time by setting τ = β

2+ it in (4.14)

Goff(t, t′) = −2ic2

∆[(Jβπ

)2cosh πt

βcosh πt′

β

]2∆, β = 2` (4.15)

The correlator looks like the product of two thermal thermal correlators and it decays intime as expected for a system that is thermalizing. Notice that the decay time is of theorder of the temperature.

It is instructive to consider the small euclidean time limit of the correlators (or large`) to obtain

Gdiag =c∆

|J(τ − τ ′)|2∆, Goff(τ, τ ′) = −2i

c2∆

|J2ττ ′|2∆(4.16)

These are the correlators in Euclidean time, close to the boundary state |B〉. We see thatwe cannot check the boundary condition (2.2) purely within the low energy limit (4.16),since the τ → 0 limit of the conformal limit of Goff gives infinity. This is not a problem,it is merely saying that in order to check the boundary condition we need to evaluate thefirst factor of Gβ(τ) in (3.12) at zero, and we should use the short distance form of theexact solution which is Gβ(0) = 1/2. This then allows us to check the boundary condition,which is of course obeyed already at the level of (3.12).

In the thermal case, we develop an emergent reparametrization symmetry that is alsoexplicitly broken by the Schwarzian action [2, 11]. We expect something similar in ourproblem. One difference is that there is a special point at τ = 0 (and τ = 2`) where theboundary sits. So we expect that the reparametrization mode, τ → ϕ(τ), should obey theboundary condition ϕ(0) = 0, ϕ(2`) = 2`. In addition, it also needs to obey ϕ′(0) = 1,ϕ′(2`) = 1. Namely, we should fix its gradient at the position of the boundary. Indeed, ifwe define

Goff, ϕ(τ, τ ′) = [ϕ′(τ)ϕ′(τ ′)]∆Goff(ϕ(τ)− ϕ(τ ′)) (4.17)

then the boundary condition at zero will be obeyed only if ϕ′(0) = 1.We know that the thermal solution spontaneously breaks the reparametrization sym-

metry to SL(2). The boundary conditions imply that only one element of SL(2) survives.Indeed, out of the three conformal Killing vectors on the circle: 1, cos τ, sin τ (for β = 2π),

4When H ∼ ψq we get ∆ = 1/q [11].

8

only the combination ζ = 1 − cos τ remains as a symmetry. Importantly, this is a sym-metry also of the off diagonal correlator Goff . The other two SL(2) generators are not asymmetry of Goff .

We mentioned above that it is useful to think of the interval τ ∈ [0, 2`] as a circle witha special point, see figure 1(b). It is useful to send this special point to infinity and map

the circle to a line. Explicitly, we map tline = −π[βJ2 tan πτ

β

]−1

. Under this map, the off

diagonal correlator becomes simply a constant,

Goff(t, t′) = −2ic2∆ (4.18)

And the surviving element of SL(2) is simply translations along this line, tline → tline+(constant).The Schwarzian action on the circle becomes

S = −αSNJ

∫dτ{tan

ϕ(τ)π

β, τ} , with {f(τ), τ} ≡

(f ′′

f ′

)′− 1

2

(f ′′

f ′

)2

(4.19)

where αS is a numerical constant [11], with boundary conditions ϕ(0) = ϕ(2`) = 0,ϕ′(0) = ϕ′(2`) = 1.

In accordance with (3.7), the one loop correction is the same as for the thermal partitionfunction. The boundary conditions for the Schwarzian variable are removing two of theSL(2) zero modes. They are still leaving one zero mode. But we should not integrate overany of these zero modes anyway, so we get the same answer.

5 Some exact diagonalization results

In this section we present some results obtained by exact diagonalization of the Hamilto-nian (2.1) for some values of N .

5.1 The pure states as a typical state in Hilbert space

First we consider the state |B〉 (2.2) for N = 30 with no further Euclidean evolution. Weexpand the state in terms of energy eigenstates as in (2.5). Up to constraints given bythe discrete symmetries, we get random looking values for cα, see figure (2). The averageenergy for this state is close to zero, as expected. This is a relatively high energy statecompared to the minimum energy state.

5.2 Correlation functions

We consider the expectation value of the operator S1(t)/2 = iψ1(t)ψ2(t) evaluated inLorentzian time on the state |B〉. See figure 3. We see that it decays to zero as we expectfor a thermalizing system. But it has small oscillations as we expect in a unitary theory. Ina unitary theory this correlator cannot decay to zero for all initial states because a suitable

9

−1.5 −1 −0.5 0 0.5 1 1.50

1

2

3

4

5

6x 10

−4

Energy / J

|ci|2

−1.5 −1 −0.5 0 0.5 1 1.5

−3.14

0

3.14

Energy / J

Ph

ase

of

c i

Figure 2: We set N = 30. In the left we plot the square of the coefficients of the non-zeroexpansion of |B〉 in terms of the energy of the energy eigenstates. (Half of the coefficientsare automatically are zero due to the (−1)F symmetry). They are random looking. Theaverage value of the square of the non-zero coefficients is 2−N/2+1, which is about 0.6×10−4.Note that the density of eigenstates is not uniform along the horizontal axis. On the rightsee the phases of the coefficients. More precisely, since the phases of the energy eigenstatescan be chosen independently for each eigenstate, we really plot the difference in phases fortwo different states |Bs〉, |Bs′〉.

linear combination of initial states should be able to give us a state with eigenvalue S1 = 1at any time. For a detailed analysis of the long time behavior in SYK see [16]. We havealso compared the answer to the twice the square of the thermal correlator, as predictedby (3.12). They match fairly closely despite the relatively low value of N .

5.3 Entanglement entropy of subsystems

Here we consider the boundary state |B〉 and we evolve it in Lorentzian time. We canorganize the Hilbert space as a tensor product of qubits, viewing the first qubit as the onewhose σ3 is given by S1 = 2iψ1ψ2, and similarly with the other qubits. More precisely,we represent the ψi in terms of qubits by a Jordan Wigner transformation5 , and then welook at the tensor decomposition of the Hilbert space in terms of the Hilbert spaces ofeach of these qubits. The initial state is simple in terms of these qubits because it obeysa condition Sk|B〉 = |B〉. It is a factorized state. However, the evolution by the SYKHamiltonian gives us general linear combinations of states with definite spins. So if weconsider the subfactor of the Hilbert space generated by the first few qubits, the initialstate is unentangled with the rest, but it will become entangled under time evolution. Infact, it becomes rapidly entangled as in a typical state of the Hilbert space [17]. By rapidly,

5Explicitly: ψ2k−1 = σ1k

∏k−1i=1 σ

3i , ψ2k = σ2

k

∏k−1i=1 σ

ik.

10

0 1 2 3 4 5 6−0.1

0

0.1

0.2

0.3

0.4

0.5

Jt

iGoff

(t,t)

2Gth

(t)2

Figure 3: We set N = 24. We plot the Lorenzian time expectation value of the operatorS1(t)/2 = iGoff(t, t) on the state |B〉 (2.4) (with ` = 0). We see that it decays over attime of the order of the interaction time 1/J . We also plot twice the square of the thermalcorrelator at β = 0. We see that (3.12) holds pretty closely despite the low value of N .

we mean a time of order 1/J , which is the characteristic interaction time. Interestingly, thetime to reach the typical entanglement is independent of the size of the subsystem6. Thisfact was demonstrated analytically at large N in [18], by showing that the density matrixfor a subset of spins factorizes at large N (see also [19, 20, 21]). In figure (4) we see aplot of the ratio of these entanglement entropies to the values we expect for a typical statein the Hilbert space, as computed in [17]. For N � 1, this typical entanglement entropyis close to maximal, SA ∼ logNA, where NA is the number qubits of the subsystem.However, at finite N the typical entanglement is slightly less than maximal. It is given bya formula Stypical(NA, N−NA), computed in [17]. We see in figure (4) that the evolution ofentanglement is fairly independent of the subsystem size and that it reaches the maximalvalue at the same time for all subsystem sizes. Similar subsytem entanglement entropieswere computed for the ground state in [22].

5.4 Euclidean correlators at finite N

Here we compute the diagonal Euclidean correlators for finite N and compare to thethermal answer, also computed at finite N . Here we are testing whether (3.11) works at

6We thank D. Stanford for emphasizing this point to us.

11

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Jt

SA

/SA

typ

ical

NA

= 1 spin

NA

= 2 spins

NA

= 4 spins

NA

= 6 spins

Figure 4: Here N = 24. Ratio of the entanglement entropy of a subfactor of NA spins (or2NA Majorana fermions), to the entanglement entropy for a typical random state in theHilbert space [17]. Something to note is that the time it takes to saturate is independentof the size of the subsystem. This is different from a local spin chain and it reflects the allto all nature of the SYK Hamiltonian.

finite N . We fix an order one value of βJ so as to have contributions from a large numberof states (we are not in the conformal limit). For low values of N , such as N = 8 we havelarge deviations of order 20 %. However, as N becomes large, we get closer results withsmaller errors, see figure (5) for examples with N = 24, 30. The way the error decreasesseems consistent with the 1/N3 scaling7.

6 Gravity interpretation

It is interesting to ask whether we can give a gravity interpretation to the above results.At the time of writing, the full bulk dual of the SYK model is unknown. However, weknow that general Nearly-AdS2 gravity or string theory has some features in common withthe low energy limit of the SYK model. In particular, in both cases we have an emergentreparametrization symmetry that is both spontaneously and explicitly broken, with anexplicit breaking given by the Schwarzian action. A simple Nearly-AdS2 model is theJackiw Teitelboim model coupled to matter, see [4, 5, 6, 7] for further discussion.

7In the left plot of (5) we are using the normalization in (3.13) where the correlator does not have to be1/2 at τ = τ ′ = 0 (at finite N). In the right we used the more natural normalization (3.9) where indeedthe correlator is 1/2 at τ = τ ′ = 0.

12

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

τ / β

Gβ(τ), βJ = 2

Gdiag

(τ,0) averaged over 150 runs

Gdiag

(τ,0) single run example

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

τ/β

Gβ(τ), βJ = 2

Gdiag

(τ,0)

Figure 5: On the left we have N = 24 and we plot the Euclidean time thermal answerand also the ratio (3.13) averaged over 150 choices of the couplings. We also ploted theratio (3.13) for one particular value of the couplings to display how it differs from theaverage. On the right we took N = 30 and we show the euclidean correlator (3.9) andthe thermal one for a single realization of the couplings. We see that as we increase N weare approaching the result (3.11). Comparing with the error for a single realization of thecouplings for N = 24 (left) and N = 30 (right) we see that it decreases as we increase N .

Here we will give a qualitative gravity picture for the solutions we had above. Moreprecisely we will discuss gravity theories with similar features (symmetries) to the onesdiscussed above. The fact that we continue to have the same diagonal correlators as thethermal state is here interpreted as saying that we continue to have the same geometry asin the Euclidean black hole. In particular, at low energies, we continue to have the sameAdS2 geometry that we had in the thermal case. In Eulidean space we have the hyperbolicdisk H2 and we imagine that there is a boundary at some finite but very large circle. Seefigure 6(a). The new feature is that we have a special point labelled by P in figure 6,where we have a kind of defect. We imagine that we have N bulk fields and that there is aboundary condition at the marked point that relates the bulk fields in pairs. For example,we can add an interaction of the form

∏k e

λSk for very large λ which kills all states exceptthe ones with eigenvalue Sk = 1 for the operators in (2.3), which are interpreted in thebulk as the product of two fermion fields iψ1ψ2 at the corresponding bulk point. We canalso imagine measuring the values of all the fields at the point P on the boundary andprojecting onto the simultaneous eigenstates of all these measurements8. In any case, weare doing a very high energy or UV-like insertion because it is localized at one point Pof the boundary circle. At all other points of the boundary circle we have the standardboundary conditions, the same as the ones we have in the thermal state. If we had bulk

8Complete measurements on an Einstein Rosen bridge were discussed in [23, 24].

13

O O1 2

(c) (d)

(b)

τ

τ=0

(a)

τ=2 = β τ=

P P

PP

t=0 t=0

H + H M

Figure 6: (a) Euclidean black hole. The geometry is the portion of H2 inside the blackboundary curve. At the point P we have special boundary condition for the bulk fields.(b) The corresponding Lorentzian black hole. It obtained by analytic continuation tolorentzian time along the line of reflection symmetry indicated by a horizontal black linepassing through P . Only the dark green region (color online) is visible to the boundaryobserver through simple measurements. The boundaries of the dark green region are thecausal horizons for such observer. The singular boundary conditions at P create a shockwave in the bulk that follows the dotted red lines. (c) By acting with operators on theboundary we can create states which are either inside or outside the horizon of figure (b).(d) By changing the boundary Hamiltonian we can render visible the whole region withinthe doted red lines.

scalar fields, we can imagine that at P , we are putting a source for the bulk scalar fields.This picture has the advantage of realizing the symmetries of the problem. In fact, we

can see this more clearly if we work in the so called poincare coordinates, where we sendthe point P to infinity. The metric can then be written as

ds2E =

dt2 + dz2

z2, ds2

L =−dt2 + dz2

z2(6.20)

14

with Euclidean or Lorentzian signature. The Lorentzian coordinates cover the whole light(and dark) green regions of figure 6(b). On this metric we want to consider a field config-uration or a boundary condition at large z that is invariant under t translations. This isbecause this was the symmetry preserved by the off diagonal low energy correlator (4.15).

We can continue this configuration to Lorentzian time in various ways. If we justcontinue t→ it in (6.20) we get a zero temperature configuration.

More interestingly, we can continue the metric along a moment of time reflection sym-metry and obtain the Rindler AdS2 coordinates, or finite temperature solution. See figure6(b). This time reflection symmetry acts as a reflection the leaves fixed the horizontal linepassing through point P in figure 6. In these coordinates the metric is the same as the onefor the thermofield double, but the fields obey different boundary conditions which breaksome of the isometries of AdS2. One important point is that we get a whole region behindthe horizon. Notice that only a portion of AdS2 is visible outside the horizon, the darkgreen region in figure 6(b).

Furthermore, inserting operators in Euclidean time, we can create more general stateson the lorentzian t = 0 slice. This gives us a precise way to generate states on this slice.Any perturbative state can be produced by a superposition of operator insertions. Inparticular, we can insert wavefunctions which are localized behind the horizon. Noticethat the map between operator insertions on the boundary and the wavefunctions on thebulk is purely kinematical, so we can formally follow the same procedure in the SYK modelto define the “bulk wavefunctions”. In the gravity picture we can localize these excitationsbehind the horizon. When we evolve in Lorentzian signature, such particles will not bevisible from the outside. One practical way we can try to see them, is to compute theexpectation value of the same field and ask whether it can become significantly large, asit would be the case with a particle that comes out the black hole and hits the boundary.This will not happen for the modes localized behind the horizon if we evolve with thelorentzian SYK hamiltonian as in figure 6(c). This suggests that for each state |Bs〉 wehave a picture which is similar to a gravity configuration with a smooth horizon. Of course,there is a shock wave some distance behind the horizon9. Though each of the states |Bs〉is special, this set of special states spans the whole Hilbert space. Each of these states isspecial because they have non-trivial correlators for the operators Sk(t). These non-trivialcorrelators decay in time as these states thermalize and become more generic, see figure 3.

6.1 Qualitative connection with other boundary state solutions

In this subsection we consider nearly-AdS2 gravity configurations containing an end of theworld brane. These are configurations that can describe pure states. The end of the world

9The distance (or time) to the shock wave becomes very small for an observer who is drops into theblack hole at very early times, at lorentzian times t � 0. For such observers the starting configurationis one that is complex and it is becoming simpler. As observed in [25] such observers see some kind offirewall.

15

“brane” is a particle in this case. We will see that by tuning its tension to high values weget a configuration that is qualitatively similar to that shown in figure 6(a).

Let us first remind the reader that a simple way to generate a gravity solution dual toa pure state is to take the eternal AdS Schwarzschild solution and perform a Z2 quotientthat exchanges the left and the right sides, a reflection along a vertical line in the Penrosediagrams, see figure 7(a,a’). Whether we can or cannot do this quotient depends on the fullgravity theory in question. In some examples that arise from string theory we can certainlydo it and the end of the world branes are the ones familiar in string constructions, see e.g.[26]. We will not discuss the full UV complete gravity theory here, we will simply considera phenomenological model where we have nearly-AdS2 gravity and we have an end of theworld particle with an arbitrary mass µ. We generate these configurations by going to acovering space containing a particle of mass µ and then doing a Z2 quotient, where theparticle sits as the Z2 invariant points. We will see that, as we increase the mass, theeffects of gravitational backreaction move this end of the world brane deeper into the leftside of the black hole geometry until we get a picture rather similar to the one in figure 6.We will now discuss this in more detail.

As explained in [10], the gravitational dynamics in nearly-AdS2 gravity can be ex-pressed in terms of the motion of a boundary particle in a rigid AdS2

10. Therefore, weare looking for a Z2 invariant configuration that contains the boundary particle and theparticle of mass µ going between two points on the boundary particle trajectory. Theboundary particle emits the massive bulk particle and it absorbes it again later, see figure7. Computing the classical solution including backreaction amounts to finding particletrajectories of this kind so that the energy momentum is conserved at the vertices. Eachof the particle trajectories is specified in terms of their SL(2) charges, and the energy mo-mentum conservation amounts to the requirement that the total SL(2) charge is zero [10].It is convenient to describe AdS2 in terms of embedding coordinates Y a = (Y −1, Y 0, Y 1),Y.Y = −(Y −1)2 − (Y 0)2 + (Y 1)2 = −1. We can also view the SL(2) charges associated tothe particle trajectories as a vector Qa. We can pick the charge for the massive particle as

Qaµ = (0, 0, µ) , Y.Q = 0 (6.21)

where we have also written the equation for the geodesic trajectory. The boundary particlesdo not follow geodesics, they move as if they where charged particles of charge q and massm in a uniform electric field in AdS2

11. Their trajectories are instead given by [10]

Y.QR = −q , Y.QL = +q , Q2 = m2 − q2 (6.22)

The trajectories specified by (6.22) look like circles in Euclidean space where the centeris at Y a ∝ Qa. The center of these circles is also the position where the future and past

10Do not confuse the boundary of AdS2 with the actual physical boundary which sits at the locationof the boundary particle. The boundary particle is in the interior of AdS2 but far away from the centralregion of interest [10].

11In terms of the parameters of the nearly-AdS2 gravity theory we have q = 2Φb = 2Φr/ε and m =q − εM , where M is the ADM mass of the black hole, and we imagine taking ε→ 0 [10].

16

(a) (b) (c)

(a’) (b’) (c’)

Figure 7: Solutions corresponding to an end of the world particle of some mass which areobtained as a Z2 quotient. (a) and (a’) Euclidean and Lorentzian black holes in nearly-AdS2 gravity. Under a Z2 quotient which is generated by a reflection along the verticalred line we get a one sided black hole and an end of the world “particle” at the red line.(a), (b), (c) correspond to situations with larger and larger values for the mass µ. (a’)(b’) (c’) are the corresponding Lorentzian solutions. The green triangle represents theregion outside the horizon. The horizontal blue doted lines represents the moment of timereflection symmetry used to connect the Euclidean and Lorentzian solutions. The red linesare the fixed points of the Z2 reflection symmetry. The black line is the trajectory of theboundary in AdS2 space.

horizons intersect in the corresponding Lorentzian black hole. For the problem we areinterested in, we expect that these circles will be displaced as in figure 7(b,c). Thereforewe make the ansatz

QaR = A(cosh ρ0, 0, sinh ρ0) , Qa

L = −A(cosh ρ0, 0,− sinh ρ0) , A2 = q2−m2 (6.23)

where we can view ρ0 as the displacement of the center of the circle relative to the origin,Y 0 = Y 1 = 0. Demanding that the sum of the charges is zero Qa

R +QaL +Qa

µ = 0 we findthat

µ = 2A sinh ρ0 (6.24)

But due to (6.22) we find that the size of each circular trajector is given by A cosh ρc = q.In order for ρc to be larger than ρ0 in (6.24) we need that

µ < 2m (6.25)

17

Which is clear if we think in terms of the balance of forces at the vertex of figure 7(c).The point is that as µ → 2m the two circles become almost tangent to each other, andafter the Z2 quotient we get a geometry qualitatively similar to that in figure 6(a). This isa large value for µ, which is comparable to the size of the UV cutoff where the boundaryparticle sits.

The analytic continuation of these configurations is such that we get a geometry thatcontains a horizon and an end of the world particle inside the horizon. The trajectoryof this particle is following a geodesic in the ambient AdS2 spacetime. As we move fromfigure 7(a’) to 7(c’) this is a geodesic that starts closer and closer to the boundary. So aswe get to figure 7(c’) it looks like a shock wave.

7 Evolution of the state under a different Hamilto-

nian

Previously we have claimed that if we start with the state |B(`)〉 at t = 0 and evolve it inLorentzian time, then we get a state that is similar to the gravity configurations in (6)(b,c)which contain a horizon and an inaccessible region behind it.

Here we will show that by evolving with a different Hamiltonian, one that is fine tunedto the particular state |Bs〉 that used to prepare the initial state, then we can get a modifiedevolution for the Schwarzian degree of freedom. Interpreted as a statement in nearly-AdS2

gravity, this modified evolution is such that it allows us to see behind the horizon. Arelated modified evolution involving a double trace deformation for pure black holes inAdS3 is being considered by [8].

This is done as follows. We add to the SYK Hamiltonian (2.1) a new term HM of theform

Htotal = HSY K + εHM , HM = −JN/2∑k=1

skiψ2k−1ψ2k (7.26)

here HM is an operator which looks like a “mass term” for the fermions. The factor of Jin HM is setting the units so that that ε is dimensionless12. It is very important for ourdiscussion that we choose the signs sk in (7.26) to be the same as the ones that describethe state |Bs(`)〉 (2.4).

At large N we could analyze this problem by solving the real time Schwinger-Dysonequations for the full Hamiltonian (7.26). This was done in [14] for a closely related situa-tion. Here we will further assume that ε is small enough so that we can first solve the SYKproblem and then treat ε as a small perturbation that will affect only the low energy dy-namics of the model. At low energies this dynamics is dominated by the reparametrizationmode f and we will consider only the effects of this mode.

12The SYK model plus a mass term was discussed in [27, 28].

18

In other words we simply evaluate the extra term, HM , on the original state andintegrate over reparametrizations

〈e−i∫dtHM (t)〉 ∼

∫DfeiS[f ]−i

∫dt〈HM (f(t))〉 ,

〈Bs(`)|HM(t)|Bs(`)〉 ∼ −NJGoff(t, t) ∼ −2NJc2

∆[Jβπ

cosh πtβ

]4∆(7.27)

We now couple the reparametrization mode by transforming Goff in this expression by areparametrization as in (4.17).

SM [ϕ] = 2εJc2∆N

∫dt

[ϕ′(t)]2∆[βJπ

cosh πϕ(t)β

]4∆= 2εJc2

∆N

∫dt(f ′)2∆ (7.28)

where

f =π

J2βtanh

πϕ(τ)

β(7.29)

By introducing a Lagrange multiplier, λ(t), it is possible to write the total action,which is the Schwarzian (4.19) plus (7.28), as

Stot = Nα

2

∫dt

[1

Jφ̇2 + λ(eφ − f ′) + ε̂Je2∆φ

], ε̂ ≡ 4c2

∆ε

α(7.30)

We set ε̂ to zero while we do the Euclidean evolution to prepare the state |Bs(`)〉 and wecan turn it on as we start the Lorentzian time evolution at t = 0. See figure 6(d). TheEuclidean time solution we are interested in is f = π

J2βtan πτ

β, which sets λ = −J . 13.

Since the equation of motion for f implies that λ is constant, we can keep the sameconstant after we start the Lorentzian evolution.

The Lorentzian evolution is simply the motion of a particle with coordinate φ on apotential

V = −λeφ − ε̂Je2∆φ = J [eφ − ε̂e2∆φ] (7.31)

This potential crosses zero at φ = φ×, given by

e(1−2∆)φ× = ε̂ , for 0 < ∆ < 1/2 (7.32)

The potential is negative for φ < φ× and positive for φ > φ×. See figure 8.Furthermore at t = 0, φ̇ = 0, since it is a moment of time reflection symmetry. And

the value of φ at t = 0 is given by

eφ0 = (f ′)t=0 =

(π

J2βtanh

πt

β

)′t=0

=π2

(βJ)2(7.33)

13The Euclidean time τ here is related to the euclidean time in previous sections via τprevious = β2 +τhere.

19

V

φφφ0 x

Figure 8: Sketch of the potential (7.31) that determines the modified evolution. If theinitial position, φ0, is less than φ×, then φ oscillates in the allowed region, given by thered line here. Since φ ∼ log z we see that z in (6.20) does not approach zero at late times.

We see that as long as φ0 < φ×, then the subsequent motion for φ remains bounded. Wewill obey this condition as long as ε is large enough

φ0 < φ× −→(π

βJ

)2(1−2∆)

< ε̂� 1 (7.34)

The last inequality comes from the condition that we can trust the reduction of the dy-namics to the reparametrization mode. We can obey both conditions in (7.34) as long asβJ � 1.

In terms of the AdS2 coordinates (6.20) we know that t ∝ f and z ∝ f ′ = eφ. Thereforewe see that the motion of the boundary along the z direction is oscillatory but bounded.In particular, it does not approach z = 0 (or φ→ −∞). This means that the region thatis visible from the boundary is the whole Poincare patch. See figure 6(d).

7.1 Adding particles behind the horizon

As we discussed above, we can add operators during the Euclidean evolution in order to addparticles to the original background. We know that we can represent any wavefunctionon top of the Hartle-Hawking vacuum by adding a suitable superposition of operatorsinserted in the lower part of the Euclidean background. Therefore, we can add particlesthat are either outside or inside the horizon, see figure 6(c). If we evolve with the originalSYK Hamiltonian, the same Hamiltonian we used to prepare the state during Euclideanevolution, then the particles behind the horizon are not visible. On the other hand, if weevolve the state with the modified Hamiltonian, then all particles become visible.

20

7.2 Relation to traversable wormholes

The addition to the Hamiltonian (7.26) is similar to the one that was considered in theproblem of traversable wormholes [9, 10]14. The boundary state |Bs〉 is obtained by mea-suring all the spins on the left half of the thermofield double, giving a result {sk}. Withthis knowledge, an observer on the right half can act with an operator that exploits theseresults, as in (7.26), and access a larger region of the spacetime. With the interactionconsidered above, the observer can access the whole region in the Poincare patch insteadof just the Rindler patch. Note that the measurement is responsible for creating the shockwave in the bulk.

8 Discussion

In this paper we have looked at a particular set of simple microstates of the SYK model.They are defined by a simple boundary condition for the Majorana fermions. They arealso joint eigenvectors of a set of commuting operators Sk. These simple states span acomplete basis of the Hilbert space of the model. We have further projected them into alower energy subspace by performing some amount ` of Euclidean time evolution.

We found that the “diagonal” correlators are exactly given by the thermal ones. Weinterpret this as saying that these simple states look completely thermalized from thepoint of view of the diagonal correlators. This also implies that the expectation values ofoperators that appear in the ψiψi OPE at order 1/N also have the same expectation valuesas in the thermal state. This is similar, in spirit, to the fact that these correlatorss for asingle value of the couplings is the same, at large N , as the average over couplings. Herethe correlators for a single state looks similar to the thermal state, which is the averageover all states. Some particular off diagonal correlators are not thermalized and know thedetails about the particular state |Bs〉. As we evolve in Lorentzian time this informationis effectively lost as the state thermalizes.

We have discussed some nearly-AdS2 phenomenological gravity theories that have simi-lar properties. In these gravity theories, the state is a configuration with only one boundarywhich contains some kind of end of the world particle in the interior. The geometry con-tains a spacetime region that is causally inaccessible from the outside. In the Lorentziansolution this end of the world particle starts in a region close to where the left boundarywould be in in the full wormhole solution and it quickly becomes a high energy shock wavewhich is at some distance behind the horizon. Observers who fall into the black hole fromthe right side at positive times experience a smooth horizon.

In the SYK model it is possible to change the Hamiltonian evolution so that the maineffects can be captured by the Schwarzian action plus a contribution induced by the extraterm in the Hamiltonian. This modification of the evolution for the reparametrizationgoldstone boson can be interpreted in the bulk as a new trajectory for the boundary. This

14In [10] the interaction was turned on only at one time. Here it is turned on for all t ≥ 0.

21

new trajectory is such that it is possible to see the whole spatial slice of the original state.In particular, we can see the region that previously was behind the horizon. This effect isbasically the same as the one that makes wormholes traversable. In fact, there is a preciseconnection between the two. In the SYK model, we can view the initial state as the onethat results from measuring a complete set of commuting operators, Sk on the left side ofa thermofield double. Then the modified evolution is essentially the same as the one thatwas considered in the context of traversable wormholes [9, 10].

It is tempting to conjecture that for more general cases, such as the black hole dual tothe D0 brane matrix model [29] (or BFSS matrix model [30]) we have a similar picture.Namely, that a full measurement in the “simple” basis of the UV state at τ = 0 on theleft side of the thermofield double state is represented by some operator which is insertedat point P in figure (6), so as to give a smooth horizon configuration. It would be nice tocheck this.

AcknowledgementsWe thank A. Almheiri, D. Stanford and Z. Yang for discussions. The thank S. Sachev

for a pre-publication copy of [14]. J.M. is supported in part by U.S. Department of Energygrant de-sc0009988. I.K. thanks the Institute for Advanced Studies for hospitality whilethis work was being finished.

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